In a lot of papers and books, I have seen the following expression of Gauss Curvature in $2$-dimensional surfaces with a conformal metric
$$\overline{g} = e^{2u}g$$
$$K - \overline{K} e^{2u} = \Delta u$$
I would like to prove it but I can't see how can i do it. I took the next expression of $K$ in isothermal parameters
$$K = -e^{-2u}\Delta u$$
and I tried to prove it but I did not obtain this expression. How can I prove it?